I use push notification (GCM) in my Android project.
According to GCM tutorial I implemented broadcast receiver and registered it in AndroidManifest.xml
In the documentation it says that :
An Android application on an Android device doesn't need to be running to receive messages. The system will wake up the Android application via Intent broadcast when the message arrives, as long as the application is set up with the proper broadcast receiver and permissions.
Check your broadcast receiver implementation and permissions.